A cylindrical conductor has uniform cross-section. Resistivity of its material increase linearly from left end to right end. If a constant current is flowing through it and at a section distance x from left end, magnitude of electric field intensity is E , which of the following graphs is correct

Options

A.

B.

C.

D.

Explanation

To solve the problem, we need to analyze the relationship between the electric field intensity \( E \) and the distance \( x \) from the left end of the cylindrical conductor, given that the resistivity \( \rho \) increases linearly along the length of the conductor.

### Step-by-step Solution:

1. **Understanding Resistivity Variation**:
The resistivity \( \rho \) of the material increases linearly from the left end to the right end. We can express this mathematically as:
\[ \rho(x) = \rho_0 + \alpha x \]
where \( \rho_0 \) is the resistivity at the left end (at \( x = 0 \)), \( \alpha \) is a constant that describes how resistivity changes with distance, and \( x \) is the distance from the left end.

2. **Using Ohm's Law**:
The electric field intensity \( E \) in a conductor is related to the current \( I \), resistivity \( \rho \), and the cross-sectional area \( A \) by the formula:
\[ E = \frac{I \cdot \rho}{A} \]
Since \( \rho \) varies with \( x \), we can substitute our expression for \( \rho(x) \) into this equation:
\[ E(x) = \frac{I \cdot (\rho_0 + \alpha x)}{A} \]

3. **Simplifying the Expression**:
We can separate the terms in the equation:
\[ E(x) = \frac{I \cdot \rho_0}{A} + \frac{I \cdot \alpha}{A} x \]
Here, \( \frac{I \cdot \rho_0}{A} \) is the y-intercept of the graph, and \( \frac{I \cdot \alpha}{A} \) is the slope of the line.

4. **Graph Interpretation**:
The equation \( E(x) = mx + c \) indicates that the graph of \( E \) versus \( x \) will be a straight line with:
- A positive slope \( m = \frac{I \cdot \alpha}{A} \)
- A y-intercept \( c = \frac{I \cdot \rho_0}{A} \)

5. **Conclusion**:
Since the resistivity increases linearly, the electric field intensity \( E \) will also increase linearly with \( x \). Therefore, the correct graph will be a straight line starting from a positive y-intercept and increasing as \( x \) increases.

### Correct Graph:
Among the options provided, the graph that represents a straight line with a positive slope and a positive y-intercept is option B.